Dynamical Borel-Cantelli lemma for recurrence theory

TL;DR

This paper extends the Borel-Cantelli lemma to dynamical systems, establishing conditions under which recurrence sets have measure zero or one, with applications to continued fractions, beta systems, and self-similar sets.

Contribution

It introduces a dynamical Borel-Cantelli lemma for recurrence sets in measure-preserving systems, under regularity conditions, linking measure to series convergence or divergence.

Findings

Recurrence sets follow a zero-full measure law based on series criteria.

Applicable to continued fractions, beta systems, and self-similar sets.

Provides a framework for understanding recurrence in dynamical systems.

Abstract

We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system $(X, \mu, T)$ with a compatible metric $d$. We prove that, under some regularity conditions, the $\mu$-measure of the following set \[ R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\N \} \] obeys a zero-full law according to the convergence or divergence of a certain series, where $\psi:\N\to\R^+$. Some of the applications of our main theorem include the continued fractions dynamical systems, the beta dynamical systems, and the homogeneous self-similar sets.